Rational Mispricing and Irrational Mispricing in Betting. Markets: Implications for Market Effi ciency Tests

Transcription

1 Rational Mispricing and Irrational Mispricing in Betting Markets: Implications for Market Effi ciency Tests Shingo Goto Toru Yamada May 2017 The views expressed here are our own and do not necessarily reflect the views of Nomura Asset Management. Any errors and inadequacies are our own. The University of Rhode Island, College of Business Administration, Ballentine Hall, 7 Lippitt Road, Kingston, RI 02881, USA. Phone: ; Fax: ; shingo_goto@uri.edu. Nomura Asset Management Co., Ltd. Dai-Ichi Edobashi Building, , Nihonbashi, Chuo-ku, Tokyo , Japan. Phone: ; Fax: ; t-yamada@nomura-am.co.jp.

2 Rational Mispricing and Irrational Mispricing in Betting Markets: Implications for Market Effi ciency Tests A Testing market effi ciency in betting markets does not necessarily get around the joint hypothesis problem because the result depends on the assumed role of bookmakers. In a simple model of monopolistic competition, bookmakers rational pricing induces the Favorite-Longshot Bias even without bettors irrationality (Rational Mispricing) and accommodates bettors irrational beliefs to exploit betting demands (Irrational Mispricing). In European football betting markets, we find significant evidence for both Rational Mispricing and Irrational Mispricing. The former mispricing, though much larger than the latter, depends on bookmakers market power and may have little to do with financial market ineffi ciency. The latter mispricing, albeit smaller, suggests that bettors overstate the persistence of team performance (Hot-Hand Bias), which may explain the reversal in betting returns and asset returns. Increased competition among bookmakers has not weakened the biases, which is diffi cult to explain only by bettors irrationality but consistent with the predicted pricing behavior of bookmakers. Keywords: Market Effi ciency, Fixed-Odds Betting, Favorite-Longshot Bias, Hot-Hand Bias, European Football (Soccer) JEL Classification: G14, G19, G29, L83 1

3 1 Introduction As the joint hypothesis problem (Fama, 1970) makes it diffi cult to address questions of market effi ciency in financial markets, numerous studies have turned to sports betting markets as a ground for testing market effi ciency. 1 Sports betting markets have an attractive feature of fixed goal-posts. That is, each bet has a well-defined termination point at which time its fundamental value is determined exogenously by the game outcomes and known with certainty, whereas the fundamental values are not directly observed in equity markets (e.g. Thaler and Ziemba, 1988). More importantly, since risks associated with game outcomes are mostly idiosyncratic and diversifiable, most predictable patterns in betting returns, if any, would be attributed to mispricing rather than to risk premiums. Consequently, systematic biases in betting prices (odds), 2 when observed, have been primarily attributed to the irrationality of the betting public (bettors in aggregate). Testing market effi ciency in sports betting markets, however, has its own challenges. In particular, the price formation process is often markedly different between sports betting markets and financial markets. Unlike market makers in financial markets, bookmakers in sports betting markets are often exposed to a substantial risk of losing money. 3 According to Levitt (2004), bookmakers can offer bets only because they are more skilled in assessing the 1 Earlier prominent studies of sports betting markets (excluding horse-race gambles) include Zuber, Gandar, and Bowers (1985), Gander, Zuber, O Brien, and Russo (1988), Sauer, Brajer, Ferris, and Marr (1988), Golec and Tamarkin (1991), Brown and Sauer (1993), Woodland and Woodland (1994), Dare and Mac- Donald (1996), Gray and Gray (1997), Gandar, Dare, Brown, and Zuber (1998), and Avery and Chevalier (1999), among others. Moskowitz (2015) has attracted renewed interests in sports betting markets. 2 A betting price is the reciprocal of the odds of the corresponding bet. We use the price and odds interchangeably when the contexts are clear. 3 Leicester City s winning of the English Premier League in was a big surprise to many. Major bookmakers like Ladbrokes Coral and William Hill offered fixed odds of 5,000-1 for the Leicester s victory at the beginning of the season, to rake in a little cash on something that would never happen. This turned into one of the largest mistakes in sports betting history. UK bookmakers lost around $15 million for the bet on the Premier League Winner. Please see Leicester s Betting Line Was a 5,000-to-1 Blunder by J. Robinson on Wall Street Journal, May 5,

4 probabilities of game outcomes than others. Through a detailed analysis of US NFL betting markets, Levitt (2004) shows that bookmakers do not necessarily balance the books, that is, they take some exposure in order to maximize their expected profits. 4 This evidence poses a significant challenge to the testing of market effi ciency in sports betting markets, because questions of market effi ciency can be addressed only when the books are balanced (Woodland and Woodland, 1994, footnote 7). In light of Levitt s (2004) insights, we take a few steps back and ask how bookmakers price-setting behavior can affect our evidence of market (in)effi ciency in sports betting markets. Suppose we find significant evidence for systematic biases in sports betting markets. Without considering the role of bookmakers, can we use the evidence to gain insights into financial market anomalies, such as momentum and reversal (value) effects in asset returns? 5 To address this question, we focus on European football (soccer) betting markets that are primarily characterized by the Fixed-Odds Betting system. In this system, betting odds reflect the amount of dollars that bookmakers commit to pay out to bettors on successful bets per unit dollar. Thus, we can view Fixed-Odds Betting markets as state-contingent claims markets which determine the value of each game outcome. However, the values of game outcomes are not necessarily determined in a competitive equilibrium. In fact, bookmakers in the Fixed Odds Betting system (e.g. European football betting markets) are likely to choose wrong prices deliberately to maximize expected profits as they are exposed to a large risk of unbalanced books. 6 As such, biased prices in Fixed-Odds Betting 4 When bookmakers balance the books, that is, when they equalize the dollar amount of payouts across different outcomes, they receive the vig (commissions, margins) with no risk exposure. 5 Momentum and value effects are widely recognized as pervasive features of financial asset prices (e.g. Asness, Moskowitz, and Pedersen, 2013), though what drive these effects remains an enduring question. Moskowitz (2015) uses value effects and return reversal effects interchangeably. 6 The concern of non-market-clearing prices is more pronounced in the Fixed Odds Betting system (e.g. European football betting markets) than in the Point-Spread Betting system (e.g. US NFL and NBA betting markets), because the books are much more diffi cult to balance in the former than in the latter. 3

5 markets may simply reflect the rational price-setting behavior of bookmakers rather than the irrational wagering behavior of the betting public. This point, however, has been largely overlooked in the voluminous literature on the effi ciency of European football betting markets. 7 To understand the price-setting decision of bookmakers, we consider a simple model of monopolistic competition among them. Using Shin s (1991,1992,1993) model as a guidance, and appealing to the one-shot nature of a Fixed-Odds Betting, we focus on a minimal setup in which a risk-neutral bookmaker solves a single-period expected profit maximization problem. Motivated by Levitt s (2004) insights, each bookmaker is assumed rational in a sense that she knows the objective probabilities of game outcomes. The bookmaker uses the knowledge to exert a market power in her specialties, but she also faces a severe competition from other bookmakers with different specialties. She cannot set the vig (commissions) freely, because bettors would simply walk away from bookmakers who charge higher vigs than others. She thus chooses her prices by maximizing her expected profits by taking the vig as given. 8 Meantime, in solving her problem, the bookmaker takes into account that her pricing decisions affect bettors demands for her bets. 9 The aggregate demand for her bets will increase in the ratio ˆp/q, where ˆp is the subjective probability of the outcome perceived by the betting public, and q is the corresponding probability implied by the betting price (odds-implied probability). This simple model allows us to draw a few 7 Examples of empirical studies on European football betting markets include Pope and Peel (1989), Cain, David and Peel (2000), Goddard and Asimakopoulos (2004), Deschamps and Gergaud (2007), Page (2009), Palomino, Renneboog, and Zhang (2009), Vlastakis, Dotsis, and Markellos (2009), Franck, Verbeek, and Neuesch (2010), Koning (2012), Nyberg (2014), Polson and Stern (2014), Feng, Polson, and Xu (2016), and Andrikogiannopoulou and Papakonstantinou (2016a,2016b,2017), among many others. 8 In Shin s (1991,1992,1993) model, two bookmakers bid for the monopoly rights to the betting market in the first stage, and the one who submits the lower vig wins the bid and offer odds in the second stage. 9 Betting demands of non-insiders in Shin s (1991,1992,1993) model are exogenous and perfectly inelastic at the balanced-book level, as his model focuses on the adverse selection problem with insiders. We consider an elastic demand, as Suits (1979) finds that price elasticities of demand in horse-race betting markets are very high considerably in excess of unity. 4

6 interesting implications about the behavior of bookmaker-determined prices. First, a bookmaker s rational pricing induces the Favorite-Longshot Bias. That is, an odds-implied probability (q) tends to overstate the chance of winning by the Longshot (that has a lower probability to win) and understate the chance of winning by the Favorite (that has a higher probability to win). This implication corroborates Shin s (1991,1992,1993) insight that bookmakers pricing decisions are at the origin of the Favorite-Longshot Bias, which is further supported by Bruce and Johnson s (2000) evidence that the Favorite- Longshot Bias exists only in bookmaker-based betting markets but not in Pari-mutuel betting markets for UK horse-races for which both forms of betting are available. Bettors misperceptions of probabilities, as suggested by the prospect theory (Tversky and Kahneman, 1992), are not suffi cient to generate the Favorite-Longshot Bias, though they can exacerbate the bias. Second, when bettors subjective probabilities are biased, bookmakers partially accommodate the biases in their pricing to exploit bettors demands. If a bookmaker were a pure monopolist and could set the vig freely, her pricing would disregard the bettors biases completely and rely only on her rational probabilistic assessments of the game outcomes. Faced with fierce competition, however, each bookmaker accommodates bettors biases partially in a manner that maximizes her expected profits under the vig constraint. 10 Third, increased competitions prompt each bookmaker to reinforce the Favorite-Longshot Bias and to distort prices to accommodate bettors biases further. Therefore, increased competitions tend to sustain, rather than suppress, biases in bookmaker-determined prices. Applying a log-linear approximation (Campbell and Shiller, 1988a,1988b) to the firstorder condition for the bookmaker s optimal price-setting, we cast our model into a tractable 10 This result is broadly consistent with Pope and Peel s (1989) and Levitt s (2004) findings that bookmakers optimally exploit bettors biases and maximize their profits by setting prices between the effi cient prices and those at which the book is balanced. Andrikogiannopoulou and Papakonstantinou (2017) note that the optimal price-setting behavior is in line with the actual practice of a bookmaker who provided their data. 5

7 multinomial logit model. We then implement the model in a sample of about 120, 000 soccer matches on Football-Data.co.uk between and Evidence for the Favorite- Longshot Bias is very strong and prevalent across leagues and over time. We also find evidence for a significant Hot-Hand Bias, but not for a Gambler s Fallacy. 12 That is, oddsimplied probabilities (q) tend to overstate the chance of winnings of teams with strong winning records, and understate the chance of winnings of teams with weak records. This evidence is broadly consistent with Andrikogiannopoulou and Papakonstantinou s (2017) finding that nearly 80% of individual bettors systematically overweigh (underweigh) teams on long winning (losing) streaks in European football betting markets. The Hot-Hand Bias generates a reversal effect (e.g. DeBondt and Thaler,1985) in betting returns, similarly to Moskowitz s (2015) finding of a value (reversal) effect in US sports betting markets, which he attributes to the bettors overreaction to information about game outcomes. 13 To complement the multinomial logit analysis, we examine the economic significance and time consistency of the two biases by constructing two hypothetical long-short portfolios. 14 The first portfolio aims to exploit the Favorite-Longshot Bias by taking a long position in bets with high prices (low odds) and a shot position in bets with low prices (high odds). The second portfolio aims to exploit the Hot-Hand Bias by taking a long position in bets on teams with weak betting returns and a short position in bets on teams with strong betting 11 We also conduct our analysis separately using a sample of about 22, 000 matches played in the Top Four European Football league divisions (England Premier League, German Bundesliga, Italian Serie A, Spanish La Liga). We use the Top Four Leagues to focus on the most active segments of the European football betting markets (similar to a Large Cap universe in stock markets). 12 The Hot-Hand Bias refers to the belief in positive autocorrelation of a non-autocorrelated random sequence, and the Gambler s Fallacy refers to the belief in negative autocorrelation of a non-autocorrelated random sequence. Our model can also capture the Home-Away Bias (over-betting on home-area teams) in the intercept term, but this term may capture other unknown effects and we refrain from discussing it. 13 Moskowitz (2015) examines how point spreads change between the Open, Close, and End (game outcome) of the betting and finds a momentum pattern between the Open and the Close, which is then reversed almost completely between the Close and the End. See also Gandar, Dare, Brown, and Zuber (1998) and Avery and Chevalier (1999) for earlier studies that examine the path of betting prices. 14 This exercise is hypothetical because one cannot short bookmakers odds in practice. 6

8 returns. Both hypothetical portfolio strategies would have produced positive long-short return spreads significantly and consistently between August 2000 and May The significant effects of the two biases do not show any signs of deterioration in the sample period despite the rapid growth and developments in European football betting markets, especially on the internet. This finding is diffi cult to reconcile with models that rely solely on bettor irrationality by ignoring the role of bookmakers, but it is consistent with the predicted price-setting behavior of bookmakers in response to increased competitions. 15 Comparing the two biases, the Favorite-Longshot Bias has been larger and more consistent than the Hot-Hand Bias, though the Hot-Hand Bias has also been significant. However, the Favorite-Longshot Bias stems primarily from a rational pricing behavior of bookmakers in the specific structure of betting markets ( Rational Mispricing ), and hence it provides little insights into financial market anomalies. The Hot-Hand Bias, on the other hand, derives from a fallacious belief of the betting public ( Irrational Mispricing ), which may have implications for the reversal in financial asset returns (e.g., Rabin and Vayanos, 2010). Still, it is worth noting that the presence and significance of the Hot-Hand Bias depend on the bookmakers rational decision to accommodate bettors biases in their prices. In sum, studies of market effi ciency in sports betting markets do not necessarily escape the joint hypothesis problem because their conclusions depend on how we account for the price-setting behavior of bookmakers. This paper demonstrates that a simple model of monopolistically competitive bookmakers can explain the prevalence and persistence of the Favorite-Longshot Bias and the Hot-Hand Bias in European football betting markets. Observed biases in bookmaker-determined prices may be largely affected by the price-setting decisions of bookmakers given a specific structure of sports betting markets, and hence may 15 Oikonomidis, Bruce, and Johnson (2015) document that a significant Favorite-Long Shot Bias prevails even in highly transparent internet-based European football betting markets. Puzzled by the finding, they conclude that markets may invariably be susceptible to fundamental risk evaluation problems. Our empirical results are consistent with theirs, but we attribute the findings to the price-setting behavior of bookmakers. 7

9 not be readily applicable to the discussion of financial market effi ciency. The rest of the paper is organized as follows: Section 2 introduces our model of Fixed- Odds Betting markets and motivates our empirical hypotheses. Section 3 presents data and empirical results. Section 4 concludes. 2 A Model of Fixed-Odds Betting Markets 2.1 Nomenclature and Notation Fixed-Odds Betting System vs. Point-Spread Betting System The traditional form of the European football (soccer) betting market is characterized by the Fixed-Odds Betting system, whereas the traditional form of the US NFL and NBA betting markets is the Point-Spread Betting system. 16 In the Point-Spread Betting system, a bookmaker offers a point spread or a handicap. 17 For example, the bookmaker quotes Home Team 2.5 and Away team This means that a bet on the Home Team will win the bet if the Home Team defeats the Away Team by 3 points or more. However, a bet on the Home Team will lose the bet even when the Home Team defeats the Away Team, if the point difference is 1 or 2 points. In a Fixed-Odds Betting for a soccer match, each bookmaker announces her odds a few days before the match is played. Bookmakers offer Fixed-Odds against each of three possible outcomes of the match, which we index by j = 1, 0, 1: the Home Team Win (j = 1), Draw 16 Fixed-Odds Betting system and Point-Spread Betting system are very different from the Pari-mutuel Betting system in which odds are proportional to quantities of money wagered. In the Pari-mutuel Betting system, bookmakers can make money from the vigs regardless of the game outcomes. 17 MLB and NHL betting markets do not use the Point-Spread Betting system because scores in baseball and ice hockey matches tend to be much lower than those in American football or basketball games. MLB and NHL betting markets employ an odds or money line that resembles the Fixed Odds Betting system, though there are a few differences in rules and conventions. See Woodland and Woodland (1994) for a study of MLB betting markets. 8

10 (j = 0), or Away Team Win (j = 1). 18 This system allows each better to make a detailed comparison between the probability of an outcome implied by the Fixed-Odds and his own assessment of the probability before making his wagering decisions Betting Odds If an odd for the j-th outcome is R j, a successful bet with a size of one yields a payoff of R j and a profit of R j 1 (when the j-th outcome obtains). R j, j = 1, 0, 1, is the outcome-contingent dividend on a unit bet on the j-th outcome. We can also view R j as the gross return (one plus return) of a bet. For instance, betting on an outcome with the odds R j = 3.3 will turn a 1 bet into 3.3, 19 implying a return of 230% if the bet is successful (the j-th outcome obtains); and into nothing ( 0) with a return of 100% if the bet is unsuccessful Betting Prices and the Vig From an investor s perspective, the market for bets in a football match corresponds to a market for contingent claims with three states of the world, j = 1, 0, 1. We can think of state-contingent claims (Arrow-Debreu securities) which pay 1 if the j-th state outcome obtains and nothing otherwise, and have their prices determined by Q i = 1/R j, j = 1, 0, 1. The bookmaker offers odds on all three states, so our analysis rules out incomplete markets. Q i is the state price for the j-th outcome. We may simply call it the betting price of the j-th outcome. The sum of the state prices of all outcomes, 1 j= 1 Q j, gives the price of a portfolio that pays 1 for sure at the end of the match (regardless of the outcome). This 18 Although more complicated bets can be placed on the score or on the half-time and full-time results, we will focus on the simplest formulation of the bet. There are papers that focus on econometric predictions of football scores (and hence game outcomes). Examples include Dixon and Coles (1997), Goddard and Asimakopoulos (2004) and Feng, Polson, and Xu (2016), though modeling football scores is outside the scope of this study. 19 This can be a Dollar, Euro, etc. 9

11 amount is larger than 1 because bookmakers charge vigs (profit margins) in offering the bets. Thus, we can write 1 j= 1 Q j = 1 + v, where v > 0 is the vig Objective, Subjective, and Odds-Implied Probabilities We can also define the normalized state prices of the three outcomes by q j = Q j /( 1 k= 1 Q k) = Q j / (1 + v), so that 1 j= 1 q j = 1, q j (0, 1) for j = 1, 0, 1. We can interpret q j as the odds-implied probability of the j-th outcome reflected in the bookmaker s odd (j = 1, 0, 1). Although several studies interpret the odds-implied probability as the market s subjective probability for each outcome, this interpretation would be possible only when the books are balanced (Woodland and Woodland, 1994). Because bookmakers can influence the implied probability q j, we distinguish it from the subjective probability of the betting public, ˆp j (0, 1), j = 1, 0, 1, 1 j= 1 ˆp j = 1. We use p o j (0, 1) to denote the objective (actual) probability of each outcome, j = 1, 0, 1, 1 j= 1 po j = 1. The ratio ˆp j/p o j captures the bias in the subjective probability ˆp j. For example, ˆp j /p o j > 1 implies that the subjective probability overstates the chance of the j-th outcome and ˆp j /p o j < 1 implies that the subjective probability understates the chance of the j-th outcome. 2.2 Biases in Betting Markets and Their Possible Relations with Financial Markets The Hot-Hand Bias and the Gambler s Fallacy Several studies suggest that investors subjective probabilities (ˆp j ) deviate from the corresponding objective probabilities (p o j ) in a systematic manner, and the biases are correlated with certain characteristics of the players/teams in the matches. According to an influential study by Gilovich, Vallone, and Tversky (1985), most individuals watching basketball believe in the Hot-Hand : they tend to believe that players who make a shot are more 10

12 likely to hit the next shot than players who miss a shot. Put differently, most individuals believe in a positive auto-correlation in basketball shots though outcomes of consecutive shots are close to independent, or slightly negatively auto-correlated (Gilovich, Vallone, and Tversky s, 1985). Camerer (1989) studies US NBA (basketball) betting markets and finds that teams with winning streaks tend to do worse than expected (by point spreads), consistent with the Hot-Hand Bias. While the Hot-Hand bias refers to the belief in positive serial-correlation of a nonserially correlated random sequence, the belief in negative serial-correlation of a non-serially correlated random sequence is called the Gambler s Fallacy (e.g. Clotfelter and Cook, 1993, Terrell, 1994). Gambler s Fallacy derives from a fallacious belief that a small sample should resemble closely the underlying population, which is often referred to as the Law of Small Numbers or Local Representativeness bias (e.g. Rabin 2002). If individuals are prone to the Hot-Hand Bias and/or the Gambler s Fallacy, both betting returns and asset returns would exhibit return reversal effects (Jagadeesh and Titman, 1993) and/or return momentum effects (DeBondt and Thaler, 1985), respectively (Rabin and Vayanos, 2010). Since risk premiums play very little role in sports betting markets, return reversal and momentum effects in sports betting markets, if any, may reflect bettors biases in assessing probabilities. Our model explicitly allows for the possibility of these biases in bettors subjective probabilities. As Camerer (1989), Rabin (2002), and Rabin and Vayanos (2010) note, the Hot-Hand Bias does not necessarily contradict with the Gambler s Fallacy, as the Hot-Hand Bias may arise from the consequence of the Gambler s Fallacy. Although individuals tend to exhibit the Law of Small Numbers (the Gambler s Fallacy) after observing a short sequence of outcomes, they tend to believe in a Hot-Hand after observing long streaks (e.g. Edwards, 1961). Rabin and Vayanos (2010) develop a model in which agents exhibit a Gambler s Fallacy in the short-run but a Hot-Hand Bias in the long-run, which they associate with 11

13 return momentum effects in a relatively short horizon and return reversal effects in a long horizon The Favorite-Longshot Bias The Favorite-Longshot Bias refers to an observation that odds-implied probabilities (q) overstate the chance of winning by the Longshot (that has a lower probability to win) and understate the winning by the Favorite (that has a higher probability to win). The following, due to Shin (1991,1992,1993), provides a precise definition of the Favorite-Longshot Bias: Definition 1 Odds-implied probabilities (q) exhibit a Favorite-Longshot Bias when q j /q i < p o j /po i if and only if po j > po i, for i, j = 1, 0, 1, i j. In stock markets, the Favorite-Longshot Bias resembles the phenomenon that stocks with greater skewness tend to have lower returns ( negative skewness premium ). 20 Not surprisingly, both the Favorite-Longshot Bias and the negative skewness premium receive similar theoretical explanations. For example, Golec and Tamarkin (1995,1998) and Garrett and Sobel. (1999) argue that the Favorite-Longshot Bias is consistent with some individuals skewness preferences. Mitton and Vorkink (2007) also emphasize the role of some investors skewness preferences in explaining the negative skewness premium in stock markets. They show that investors with greater demand for positive skewness will consciously choose to remain under-diversified because diversification is a two-edged sword: it eliminates undesired variance in return distributions, but also eliminates desired skewness (p.1256). Julien and Selanie (2000) and Snowbert and Wolfers (2010) argue that the Favorite- Longshot Bias is consistent with bettors misperceptions of probabilities, as suggested by the prospect theory (Kahneman and Tversky, 1979, Tversky and Kahneman, 1992).and 20 See Mitton and Vorkink (2007), Brunnermeier, Gollier, and Park (2007), Barberis and Huang (2008), and Amaya, Christoffersen, Jacobs, and Vasquez (2015), among others. 12

14 implemented with a probability weighting function. In explaining the negative skewness premium, Barberis and Huang (2008) develop a model with cumulative prospecting investors who are more unhappy for downside risk than happy about upside potential. They show that, under the cumulative prospect theory preferences of Tversky and Kahneman (1992), investors transform objective probabilities using a weighting function that overweighs the tails of the probability distribution, and this causes positively skewed securities to become over-priced and to earn negative average risk-adjusted returns. From a similar but different angle, Brunnermeier, Gollier, and Park (2007) argue that investors tend to overstate the probabilities of large upside potential, because a small optimistic bias in beliefs typically leads to first-order gains in anticipatory utility and only second-order costs in realized outcomes (p.1092). In their model, investors optimize over beliefs of outcomes, as opposed to taking probabilities as given (Brunnermeier and Parker, 2005). This optimizing behavior of investors leads to, among other predictions, a strong preference for stocks with positively skewed distributions. This argument naturally extends to both the Favorite-Longshot Bias and the negative skewness premium. In our model, these theoretical effects affect bookmaker-determined prices though biases in bettors subjective probabilities (ˆp j ) and betting demands both of which are described below. In light of recent evidence on bettors; misperceptions of probabilities, our analysis will incorporate a probability weighting function that is suggested by Kahneman and Tversky (1992) and Camerer and Ho (1994). 2.3 Bookmakers Risk Exposure Bookmakers in the Fixed-Odds Betting and Point-Spread Betting systems can be exposed to a risk of losing money when the books do not balance. In the Point-Spread Betting system, a bookmaker can seek a point spread to balance the book, i.e., to balance the dollar amount wagered on each outcome. If the books are balanced, bookmakers can make money 13

15 regardless of the final game outcomes. We can thus view the market clearing spread as an expression of the dollar-weighted average opinion of the betting public (Camerer, 1989). In practice, bookmakers may choose wrong spreads to take speculative positions when they have superior information (Levitt, 2004). Bookmakers in the Fixed-Odds Betting system are in different situations. They can have large exposures to the risk of losing money, because odds are set before the match and are not adjusted to betting volumes. By way of example, let s consider a match in which there are only two outcomes, H (Home team win) and A (Away team win). To simplify matters, we ignore the vig (i.e., v = 0) and consider actuarially fair gambles. Suppose that 2/3 of the betting public (in terms of the quantity of money) bets on the outcome H and the remaining 1/3 bets on the outcome A. In this case, the actuarially fair odds are 1.5 for H and 3.0 for A. (H is the Favorite and A is the Longshot in this example.) Now suppose that the bookmaker fixes and announces the actuarially fair odds at 1.5 for H and 3.0 for A. In order for the bookmaker s book to balance, 2/3 of the wagers should go to H and 1/3 of the wagers should go to A. In this case, 1.5 2/3 and 3 1/3 always equal one, thus the bookmaker has no risk exposure. However, this balance hardly obtains because both bets have exactly the same expected return. Risk neutral bettors would be indifferent between the two bets, and hence it is more likely that similar amounts of wagers are placed on the two outcomes rather than they are allocated according to the respective probabilities (2/3 for H and 1/3 for A). When equal dollar amounts are wagered on both outcomes, bookmakers would lose money when the outcome A obtains (1 3 1/2 = 0.50), while they can earn profits when the outcome H obtains ( /2 = +0.25). As this simple example illustrates, bookmakers in Fixed-Odds Betting markets may end up taking large exposures to game outcomes even when they offered actuarially fair gambles. 14

16 2.4 Model Setup Bettors Subjective Probabilities The majority of existing studies do not distinguish between the implied probability q j and the subjective probability ˆp j. However, we explicitly consider possible biases in the subjective probability of the betting public (ˆp j ) and examine how the biases affect the implied probability (q j ). We consider two sources that can distort the subjective probability (ˆp j ) from the objective one (p o j ). The first source is the Hot-Hand Bias or the Gambler s Fallacy. We can incorporate the Hot-Hand Bias or the Gambler s Fallacy as follows. Let X j be the recent winning records of the Home team (j = 1) or the Away team (j = 1). We set X 0 = 0 for the Draw case. We then consider the following tractable specification of the subjective probability: ln ˆp j p j = X j δ + const j, for j = 1, 0, 1, (1) where p j is a reference probability which may be subject to the probability weighting in the prospect theory, as we describe below. const j is a constant term that plays no significant role in our analysis but ensures the subjective probabilities of the three outcomes to sum to one. X 1 (X 1 ) is higher when the Home (Away) team has a stronger winning record in recent matches. The Hot-Hand Bias implies δ > 0 in (1), and the Gambler s Fallacy implies δ < 0 for j = 1 and j = 1. The sign of δ may depend on the length of winning records in the measurement of X j, as the literature suggests that the Gambler s Fallacy arises in the short run but the Hot-Hand Bias becomes dominant in the long-run (e.g. Rabin, 2002, Rabin and Vayanos, 2010). Although we focus on the Hot-Hand Bias and the Gambler s Fallacy, due to their close association with the return reversal and momentum effects in financial markets, we can include a set of other team characteristics in X j when we hypothesize them to affect the subjective probability of the betting public. The second source of the bias in the subjective probability is the misperception of prob- 15

17 abilities as summarized by the probability weighting function in the prospect theory of Tversky and Kahneman (1992). Recent studies demonstrate the relevance of the prospect theory, especially the significance of probability weighting, in characterizing the wagering behavior of bettors (e.g. Julien and Selanie, 2000, Snowbert and Wolfers, 2010, Andrikogiannopoulou and Papakonstantinou, 2016a,2016b,2017). To model the probability weighting, we assume that the reference probability p j takes the following form, which is suggested by Tversky and Kahneman (1992) and Camerer and Ho (1994). 21 p j = (p o j )1 ε [ 1 ] 1, j = 1, 0, 1. (2) i= 1 (po i )1 ε 1 ε The prospect theory typically assumes ε [0, 1] to promote the idea that bettors tend to overreact to small probability events, but underreact to large probabilities (i.e., the Favorite-Longshot Bias). When ε = 0, p j is the same as the objective probability p o j. A larger deviation of ε from zero toward one implies a larger distortion of p j from p o j. The two equations (1) and (2) imply the following relation between a log odds-ratio of the subjective probabilities of the betting public ( ˆp) and that of the objective probabilities (p j ): ln ˆp j ˆp i = (1 ε) ln po j p o i + (X j X i ) δ + const, for i, j = 1, 0, 1, i j, (3) where const does not play an important role. In (3), δ > 0 (or δ > 0) captures the Hot- Hand Bias (or the Gambler s Fallacy). ε > 0 is consistent with the Favorite-Longshot Bias, according to the definition in Section Barberis and Huang (2008) consider a two-parameter asymmetric probability weighting function in a model with cumulative prospect theory, but this study focuses on a single parameter case. 16

18 2.4.2 Betting Demands Bettors subjective probabilities affect the aggregate demand. We consider the following specification of the aggregate betting demand for the j-th outcome (j = 1, 0, 1). ( ) η ˆpj x j (ˆp j, q j ) = for j = 1, 0, 1, (4) q j where η > 0 is the elasticity of demand with respect to the betting price per perceived chance, q/ ˆp. Suits (1979) reports that the elasticity η is in excess of unity in horserace betting markets. The form of the demand function (4) appears frequently in the macroeconomics literature with monopolistic competition (e.g., Dixit and Stiglitz, 1977). It describes that bettors tend to wager more (less) when expected return of the bet, according to their subjective probabilities, ˆp j R j = ˆp j /Q j, is higher. The benchmark case is when the implied probability and the subjective probability coincide, i.e., when q j = ˆp j. In this case, we assume that the aggregate amount wagered on the bet is one. The demand function (4) is a reduced-form specification of aggregate betting demands that is consistent with the demand of a representative bettor who maximizes 1 j= 1 (ˆp jx j R j 1 ρ xρ j ) ignoring the vig, where ρ = η > 1. This bettor maximizes his anticipated payoff (ignoring the vig), 1 j= 1 ˆp jx j R j, after subtracting a convex betting cost, 1 j= 1 xρ j, ρ > 1, that penalizes concentrated bets on particular game outcomes. We are not proposing this particular objective function as a model of bettors wagering decisions, but we can give a simple economic interpretation to the aggregate demand function (4). Being a reducedform demand function, it is not meant to model interactions of bettors with different risk preferences, heterogeneous probabilistic beliefs, and varying budgets. As Manski (2006) notes, however, aggregate demands should depend on the joint distribution of preferences, beliefs, and budgets across agents. For example, the literature has suggested locally convex utilities, skewness preferences, prospect theories with probability weighting functions, etc. to explain bettors wagering decisions, while Ali (1977) and 17

19 Gandhi and Serrano-Padial (2015) stress the importance of the heterogeneity in bettors subjective probabilities. The list can go on, but the literature is yet to agree on a standard model that can be used to aggregate individual bettors wagering decisions to generate a tractable aggregate betting demand. 22 Therefore, this study assumes a tractable singleparameter aggregate betting demand that allows us to focus on the price-setting behavior of monopolistically competitive bookmakers. Specifically, the single parameter η captures the elasticity of the aggregate betting demand. The elasticity parameter η increases when different bets (e.g. bets on Real Madrid, Bayern Münich, Chelsea, Juventus, etc.) become closer substitutes, i.e., when each bet becomes closer to financial securities. We expect η to increase with the institutional and technological developments of the markets and with the number of professional bettors and information service providers. Thus, we take the liberty of interpreting η as a measure of market sophistication in sports betting markets. In perfectly competitive securities markets, η tends to infinity The Bookmaker s Problem Each bookmaker is more skilled in assessing the probability of each game outcome than others (Levitt, 2004). In our model, the bookmaker is rational in a sense that she employs the objective probability p o j (j = 1, 0, 1) in setting her odds for each outcome. Since risks associated with betting outcomes are idiosyncratic and diversifiable across matches, we assume that each bookmaker is risk-neutral, similarly to Shin (1991,1992,1993). The risk-neutral bookmaker s problem is to maximize her expected profit by setting Q j 22 More recently, Andrikogiannopoulou and Papakonstantinou (2016a,2016b,2017) analyze a unique dataset of wagering activities by 500 randomly selected individuals in European sports betting markets. Their studies point to the relevance of the prospect theory, especially the significance of probability weighting, in characterizing bettors risk preferences and betting demands. They abstract away from bookmakers pricesetting behavior, however. Thus, their focus on individuals betting demands and our focus on bookmakers price-setting decisions are complementary. 18

20 for the three possible outcomes (j = 1, 0, 1): max E[Π] = {Q j } j=1,0, 1 1 j= 1 x j (ˆp j, q j ) 1 j= 1 p o jx j (ˆp j, q j ) 1 Q j, (5) subject to the vig constraint, 1 j= 1 Q j = 1 + v, where ˆp j (0, 1), p o j (0, 1), and Q j (0, 1 + v). The demand function x j (ˆp j, q j ) is given in (4) with q j = Q j / (1 + v). We focus on the interior solution and preclude the unrealistic cases of q j = 0 (R j ) or q j = 1 (R j = 1/ (1 + v) < 1) Monopolistic Competition among Bookmakers If the bookmaker had no vig constraint, the unconstrained solution to her optimization problem would be Q j = (1 + 1 η )po j, j = 1, 0, 1, implying 1 j= 1 Q j = η. Thus the optimal vig (markup) of the monopolist would be v = 1 η. Q j (j = 1, 0, 1) is the optimal price if the bookmaker were a pure monopolist who could set the vig freely. In practice, each bookmaker can exert her market power only within her specialized segment, and the vig is determined by competitive forces in the bookmaking market. Since bettors are free to walk away from high-vig betting opportunities for low-vig betting opportunities, each bookmaker has to take the level of vig (v < 1 η ) as given. We introduce θ = 1 ηv as a measure of the degree of monopolistic competition among bookmakers. Note that v = (1 θ) /η = (1 θ) v where v is the vig level chosen by a pure monopolist. Pure monopoly would imply θ = 0, but the vig level will decrease with θ. That is, a higher value of θ (0, 1) implies that more intense competition among bookmakers. With the monopolistic competition, the bookmaker has to offer betting prices that are ) lower than the monopolistic prices, and hence Q j < Q j or q j/p o j (1 < + 1 η / (1 + v) for j = 1, 0, 1. 19

21 2.5 Model Implications The first-order condition for the optimality given the constraint 1 j= 1 Q j = 1 + v (or 1 j= 1 q j = 1) and possible biases in bettors subjective probabilities (3) implies the following identity expressed in terms of odds ratios : 23 η ln ˆp j (1 + η) ln q 1+ 1 η p o j j 1+v q + ln j 1 = 0, for i, j = 1, 0, 1, i j. (6) ˆp i q i 1+ 1 η p o i 1+v q i 1 The assumption that the bookmaker has to offer betting prices lower than the monopolistic ) prices (i.e., Q j < Q j and hence q j/p o j (1 < + 1 η / (1 + v)) ensures that the term inside the square bracket in (6) is positive. This condition also satisfies the second-order condition, ) q j /p o j (1 < + 2 η / (1 + v) for j = 1, 0, 1. The first-order condition (6) is nonlinear and diffi cult to interpret. To obtain useful empirical predictions, we apply the log-linear approximation of Campbell and Shiller (1988a,1988b) around p o j /q j = 1 for j = 1, 0, 1. Proposition 1 Let p o j be the objective probability of the j-th game outcome, j = 1, 0, 1, which the bookmaker uses. Let ˆp j denote the subjective probability of the betting public for the same outcome, where the log odds ratio ln ˆp j ˆp i is characterized by the relation (3). Then, solutions to the bookmaker s constrained optimization problem (5) should satisfy the following approximate identity: where ln q j q i γ ln po j p o i + (X j X i ) δφ + const, for i, j = 1, 0, 1, i j, (7) θ (1 + ηε) γ = 1 + η + ηθ(1 ε), (8) ηθ φ = (1 + η) (1 + θ) (9) with θ = 1 ηv (0, 1) capturing the degree of monopolistic competition. In (7), const is a constant term that plays no significant role in our analysis. 23 Odds ratio here is a statistical/econometric terminology referring to the ratio of two probabilities. The meaning of odds here is different from betting odds. 20

22 In the case of a pure monopoly (θ = 0), both γ and φ become zero and the odds-implied probability q j would reveal the bookmaker s superior assessment of the objective probability p o j fully for j = 1, 0, 1. That is, when the bookmaker could set the vig freely to maximize her expected profits, her pricing would reveal her superior probabilistic assessments of the game outcomes, regardless of the Hot-Hand Bias, the Gambler s Fallacy, or the probability weighting. In another extreme case of perfectly inelastic betting demand (i.e., η 0), θ tends to one and hence we would have γ = 1 and φ = 0. That is, a square root pricing rule that resembles Shin s (1991) obtains. This would not be surprising because Shin (1991) assumes exogenous and perfectly inelastic betting demands. In practice, aggregate betting demands are elastic (η > 0), and bookmakers face competition from other bookmakers and potential entrants, θ (0, 1). In this practical setting, the log approximate identity (7) yields a few interesting implications: First, γ is positive, meaning that a Favorite-Longshot Bias obtains according to the definition in Section The Favorite-Longshot Bias (γ > 0) arises even in the absence of bettor irrationality (δ = 0 and ε = 0). The Favorite-Longshot Bias is thus driven by the bookmaker s optimal price setting behavior under monopolistic competition, and it represents the Rational Mispricing in sports betting markets. Our model implies that bookmakers pricing decisions are at the origin of the Favorite-Longshot Bias. This implication is similar to Shin s (1991,1992,1993) result, but our model does not assume the presence of insiders (bettors who have inside information about the game outcomes). This implication is also consistent with the empirical evidence of Bruce and Johnson (2000) who show that the Favorite-Longshot Bias exists only in bookmaker-based betting markets but not in Pari-mutuel betting markets for UK horse-races for which both forms of betting are available. We also note that γ increases with ε, meaning that bettors misperceptions of probabilities, along the line of the prospect theory (Tversky and Kahneman, 1992), can exacerbate 21

23 the Favorite-Longshot Bias. However, the probability weighting alone does not lead to the Favorite-Longshot Bias. Monopolistic competition among bookmakers (θ > 0) is necessary and the Favorite-Longshot Bias (γ > 0) is consistent with the rational price-setting behavior of the bookmakers. A higher demand elasticity (η) and a larger misperception of probabilities (ε) tend to make the Favorite-Longshot Bias larger. Second, even when bettors subjective probabilities exhibit biases (e.g. the Hot-Hand Bias and/or the Gambler s Fallacy), the bookmaker s optimal price setting eliminates more than half of the biases, as φ (0, 1 2 ). However, the bookmaker will not correct the biases completely, because she can take advantage of bettors demands by accommodating their biases in order to maximize her expected profits. Thus, Irrational Mispricing in bookmaker-determined prices arise from bettor s biases. Third, for a given level of elasticity (or sophistication of the betting market) η, the degree of Rational Mispricing (γ) increases with the degree of monopolistic competition among bookmakers, θ. Also for a given level of elasticity η and bettors biases δ, the degree of Irrational Mispricing (δφ) increases with the degree of monopolistic competition, θ, as φ is increasing in θ. That is, as the competition among bookmakers gets more intense, the degree of Rational Mispricing and that of Irrational Mispricing increase, rather than decrease. Fourth, for a given level of θ, an increase in the elasticity of aggregate demands η, or the sophistication of the betting market, reduces the vig and the degree of the Favorite-Longshot Bias γ. This implication is consistent with the notion that each bookmaker s market power in her specialty declines as η increases, i.e., as the betting market gets closer to competitive securities markets. However, the relation between the vig and the Favorite-Longshot Bias is ambiguous, as a decline in the vig can be associated with either an increase in η or an increase in θ. That is, the Favorite-Longshot Bias (Rational Mispricing) does not necessarily decrease with a decline in the vig. On the other hand, φ increases with η, but δ is likely to 22

24 decrease with η. Thus, the relation between the betting market s sophistication (η) and the degree of Irrational Mispricing (δφ) is ambiguous. Even when the betting market gets more sophisticated, the degree of Irrational Mispricing does not necessarily decline, contrary to what we tend to anticipate. Among these four implications, the first two give unambiguous empirical predictions, and we cast these into empirical hypotheses below. The remaining two implications, while surprising and somewhat counter-intuitive, are diffi cult to test because we cannot observe the degree of competition among bookmakers (θ), that of market sophistication (η), or that of bettors biases (δ) directly. We only observe the vig, v = 1 θ η, which is decreasing in both θ and η. We thus ask how the recent growth in sports betting markets, as manifested by a steady decline in the vig, has been associated with the level of Rational Mispricing (γ) and that of Irrational Mispricing (δφ) in European football betting markets. 2.6 Empirical Hypotheses Main Hypotheses We can cast (7) naturally into a multinomial logit regression model, 24 which can be implemented in data without any ad hoc modifications. Specifically, using the Draw outcome (j = 0) as the pivot category and setting X 0 = 0, we can express the log odds ratios of the Home team win (j = 1) and the Away team win (j = 1) as: ln po 1 p o 0 = (1 + γ) ln q 1 q 0 X 1 β + a 1, ln po 1 p o 0 = (1 + γ) ln q 1 q 0 X 1 β + a 1, (10) where γ is given by equation (8) and β = δηθ 1 + η + ηθ (1 ε). 24 Former studies that use multinomial logit models in betting markets include Figlewski (1979), Vlastakis, Dotsis, Markellos (2009), and Nyberg (2014), among others. 23

25 a 1 and a 1 are constants whose difference may capture the Home-Away Bias. But we refrain from pushing this interpretation as these intercept terms may capture other effects. We implement the empirical model (10) with maximum likelihood to test the following hypotheses: Hypothesis 1 (Rational Mispricing) Bookmakers price-setting behavior in monopolistic competition induces a Favorite-Longshot Bias, γ > 0. The null hypothesis is γ = 0. Hypothesis 2 (Irrational Mispricing) The team s past performance can bias the subjective probabilities of the betting public and affects the odds-implied probabilities, after controlling for the Favorite-Longshot Bias, and hence β 0. β > 0 implies a Hot-Hand Bias whereas β < 0 implies a Gambler s Fallacy. The null hypothesis is β = 0. We use γ as a measure of Rational Mispricing because the Favorite-Longshot Bias arises from the bookmakers price setting behavior, even in the absence of bettors biases. Rational Mispricing is specific to the organizational structure of sports betting markets, and hence is not transferable to financial markets. The probability weighting, as suggested by the prospect theory (Tversky and Kahneman, 1992), can exacerbate the Favorite-Longshot Bias as γ increases with ε. However, the probability weighting ε > 0 is not suffi cient for the Favorite-Longshot Bias (γ > 0) to arise. Monopolistic competition among bookmakers θ > 0 plays a key role in the generation of the bias. On the other hand, we associate β with Irrational Mispricing as it is originated from biases in bettors subjective probabilities. For example, a return reversal (value) effects associated with the Hot-Hand Bias, if any, may help us understand similar effects in financial markets (Rabin and Vayanos, 2010). The probability weighting ε > 0 can again intensify the observed degree of Irrational Mispricing, as the absolute value of β increases with ε. 24